{"paper":{"title":"Sharp optimal recovery in the two-component Gaussian Mixture Model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Mohamed Ndaoud","submitted_at":"2018-12-19T16:53:12Z","abstract_excerpt":"This paper studies the problem of clustering in the two-component Gaussian mixture model where the centers are separated by $2\\Delta$ for some $\\Delta>0$. We characterize the exact phase transition threshold, given by $$ \\bar{\\Delta}_n^{2} = \\sigma^{2}\\left(1 + \\sqrt{1+\\frac{2p}{n\\log{n}}} \\right)\\log{n}, $$ such that perfect recovery of the communities is possible with high probability if $\\Delta\\ge(1+\\varepsilon)\\bar \\Delta_n$, and impossible if $\\Delta\\le (1-\\varepsilon)\\bar \\Delta_n$ for any constant $\\varepsilon>0$. This implies an elbow effect at a critical dimension $p^{*}=n\\log{n}$.\n  "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08078","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1812.08078/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}